Report from Dagstuhl Seminar 18421 Algorithmic Enumeration: Output-sensitive, Input-Sensitive, Parameterized, Approximative Edited by Henning Fernau1, Petr A. Golovach2, and Marie-France Sagot3 1 Universität Trier, DE, [email protected] 2 University of Bergen, NO, [email protected] 3 University Claude Bernard – Lyon, FR, [email protected] Abstract This report documents the program and the outcomes of Dagstuhl Seminar 18421 “Algorithmic Enumeration: Output-sensitive, Input-Sensitive, Parameterized, Approximative”. Enumeration problems require to list all wanted objects of the input as, e.g., particular subsets of the vertex or edge set of a given graph or particular satisfying assignments of logical expressions. Enumeration problems arise in a natural way in various fields of Computer Science, as, e.g., Artificial Intelligence and Data Mining, in Natural Sciences Engineering, Social Sciences, and Biology. The main challenge of the area of enumeration problems is that, contrary to decision, optimization and even counting problems, the output length of an enumeration problem is often exponential in the size of the input and cannot be neglected. This makes enumeration problems more challenging than many other types of algorithmic problems and demands development of specific techniques. The principal goals of our Dagstuhl seminar were to increase the visibility of algorithmic enumeration within (Theoretical) Computer Science and to contribute to establishing it as an area of “Algorithms and Complexity”. The seminar brought together researchers within the algorithms community, other fields of Computer Science and Computer Engineering, as well as researchers working on enumeration problems in other application areas, in particular Biology. The aim was to accelerate developments and discus new directions including algorithmic tools and hardness proofs. Seminar October 14–19, 2018 – http://www.dagstuhl.de/18421 2012 ACM Subject Classification Mathematics of computing → Enumeration, Mathematics of computing → Combinatorial algorithms, Theory of computation → Design and analysis of algorithms, Theory of computation → Complexity classes Keywords and phrases algorithms, input-sensitive enumeration, output-sensitive enumeration Digital Object Identifier 10.4230/DagRep.8.10.63 Edited in cooperation with Benjamin Gras (University of Orleans, FR) Except where otherwise noted, content of this report is licensed under a Creative Commons BY 3.0 Unported license Algorithmic Enumeration: Output-sensitive, Input-Sensitive, Parameterized, Approximative, Dagstuhl Reports, Vol. 8, Issue 10, pp. 63–86 Editors: Henning Fernau, Petr A. Golovach, and Marie-France Sagot Dagstuhl Reports Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 64 18421 – Algorithmic Enumeration 1 Executive Summary Marie-France Sagot (University Claude Bernard – Lyon, FR) Dieter Kratsch (Université de Lorraine, FR) Henning Fernau (Universität Trier, DE) Petr A. Golovach (University of Bergen, NO) License Creative Commons BY 3.0 Unported license © Marie-France Sagot, Dieter Kratsch, Henning Fernau, and Petr A. Golovach About fifty years ago, NP-completeness became the lens through which Computer Science views computationally hard (decision and optimization) problems. In the last decades various new approaches to solve NP-hard problems exactly have attracted a lot of attention, among them parameterized and exact exponential-time algorithms, typically dealing with decision and optimization problems. While optimization is ubiquitious in computer science and many application areas, relat- ively little is known about enumeration within the “Algorithms and Complexity” community. Fortunately there has been important algorithmic research dedicated to enumeration prob- lems in various fields of Computer Science, as, e.g., Artificial Intelligence and Data Mining, in Natural Sciences Engineering and Social Sciences. Enumeration problems require to list all wanted objects of the input as, e.g., particular subsets of the vertex or edge set of a given graph or particular satisfying assignments of logical expressions. Contrary to decision, optimization and even counting problems, the output length of enumeration problems is often exponential in the size of the input and cannot be neglected. This motivates the classical approach in enumeration, now called output-sensitive, which measures running time in (input and) output length, and asks for output-polynomial algorithms and algorithms of polynomial delay. This approach has been studied since a long time and has produced its own important open questions, among them the question whether the minimal transversals of a hypergraph can be enumerated in output-polynomial time. This longstanding and challenging question has triggered a lot of research. It is open for more than fifty years and most likely the best known open problem in algorithmic enumeration. Recently as a natural extension on research in exact exponential-time algorithms, a new approach, called input-sensitive, which measures the running time in the input length, has found growing interest. Due to the number of objects to enumerate (in the worst case), the corresponding algorithms have exponential running time. So far branching algorithms are a major tool. Input-sensitive enumeration is strongly related to lower and upper combinatorial bounds on the maximum number of objects to be enumerated for an input of given size. Such bounds can be achieved via input-sensitive enumeration algorithms but also by the use of combinatorial (non-algorithmic) means. The area of algorithmic enumeration is in a nascent state, though it has a huge potential due to theoretical challenges and practical applications. While output-sensitive enumeration has a long history, input-sensitive enumeration has been initiated only recently. Natural and promising approaches like using parameterized or approximative approaches have not been explored yet in their full capacities. The principal goals of our Dagstuhl seminar were to increase the visibility of algorithmic enumeration within (Theoretical) Computer Science and to contribute to establishing it as an area of “Algorithms and Complexity”. The seminar brought together researchers within the algorithms community, other fields of Computer Science and Computer Engineering, as well as researchers working on enumeration problems in other application areas, in particular, in Biology. Besides the people already working with enumeration, researchers from other Henning Fernau, Petr A. Golovach, and Marie-France Sagot 65 fields of Computer Science were invited. In particular, researchers who are interested in Parameterized Complexity and different aspects of counting problems were participating in the seminar. The aim was to accelerate developments and discuss new directions including algorithmic tools and hardness proofs. The seminar collected 44 participants from 13 countries. The participants presented their recent results in 18 invited and contributed talks. Open problems were discussed in several open problem and discussion sessions. 1 8 4 2 1 66 18421 – Algorithmic Enumeration 2 Table of Contents Executive Summary Marie-France Sagot, Dieter Kratsch, Henning Fernau, and Petr A. Golovach . 64 Overview of Talks Modular counting of directed Hamiltonian cycles by enumerating solutions to quadratic equations Andreas Björklund .................................... 68 Ideal-preferred Enumeration of Minimal Dominating Sets in Graphs Oscar Defrain ....................................... 68 Enumerating Vertices of Covering Polyhedra with Totally Unimodular Constraint Matrices Khaled M. Elbassioni and Kazuhisa Makino ...................... 69 The Minimal Extension of a Partial Solution Henning Fernau ...................................... 69 Enumeration of Preferred Extensions in Almost Oriented Digraphs Serge Gaspers ....................................... 70 Listing All Maximal k-Plexes in Temporal Graphs Anne-Sophie Himmel ................................... 70 Constructive and nonconstructive enumeration of designs Petteri Kaski ....................................... 71 Node Similarity with q-Grams for Real-World Labeled Networks Andrea Marino ...................................... 71 On Solving Enumeration Problems with SAT Oracles João Marques-Silva .................................... 72 Minimal Connected Dominating Sets below 2n Michal Pilipczuk ..................................... 72 Enumerating minimal dominating sets in triangle-free graphs Jean-Florent Raymond .................................. 73 The Maximum Number of Minimal Dominating Sets in a Tree Günter Rote ........................................ 73 Efficiently Enumerating Hitting Sets of Hypergraphs Arising in Data Profiling Martin Schirneck ..................................... 73 A panorama of enumeration complexity Yann Strozecki ...................................... 74 How to use the Solutions of Enumeration in Practice Takeaki Uno ........................................ 74 A Complexity Theory for Hard Enumeration Problems Heribert Vollmer ..................................... 75 Open problems The DIVERSE X Paradigm Michael R. Fellows and Frances Rosamond ...................... 75 Henning Fernau, Petr A. Golovach, and Marie-France Sagot 67 Various enumeration perspectives Henning Fernau ...................................... 78 Enumeration of tree width-t-modulators Fedor V. Fomin and Saket Saurabh ........................... 79 Minimal separators in graphs Serge Gaspers